Question

the exponential models describe the population of the indicated country, a, in millions, t years after 2006. which country has the greatest growth rate? by what percentage is the population of that country increasing each year? country 1: $a = 149.7e^{-0.002t}$ country 2: $a = 132.9e^{0.004t}$ country 3: $a = 1097.7e^{0.012t}$ country 4: $a = 28.9e^{0.026t}$ country $square$ has the greatest growth rate.

Explanation:

Step1: Recall exponential growth model

The exponential growth model is \( A = A_0e^{kt} \), where \( k \) is the growth rate (if \( k>0 \)) or decay rate (if \( k < 0 \)). To find the country with the greatest growth rate, we compare the values of \( k \) for each country.

Step2: Identify \( k \) for each country

• Country 1: \( A = 149.7e^{- 0.002t} \), so \( k=- 0.002 \) (decay, since \( k<0 \))
• Country 2: \( A = 132.9e^{0.004t} \), so \( k = 0.004 \)
• Country 3: \( A=1097.7e^{0.012t} \), so \( k = 0.012 \)
• Country 4: \( A = 28.9e^{0.026t} \), so \( k=0.026 \)

Step3: Compare the \( k \) values

We compare \( 0.004 \), \( 0.012 \), and \( 0.026 \) (we ignore the negative \( k \) for growth rate comparison). Among \( 0.004 \), \( 0.012 \), and \( 0.026 \), \( 0.026 \) is the largest, which corresponds to Country 4.

Step4: Find the percentage growth rate

For the exponential growth model \( A = A_0e^{kt} \), the percentage growth rate is found by using the formula for exponential growth in the form \( A=A_0(1 + r)^t \), and we know that \( e^{k}=1 + r \), so \( r=e^{k}-1 \). For Country 4, \( k = 0.026 \).

Calculate \( r=e^{0.026}-1 \). Using a calculator, \( e^{0.026}\approx1.02635 \), so \( r\approx1.02635 - 1=0.02635 \), which is approximately \( 2.635\% \) (or we can also note that for small \( k \), \( e^{k}\approx1 + k+\frac{k^{2}}{2}+\cdots \), so \( r\approx k \) in percentage terms, but the exact way is \( r = e^{k}-1 \)).

First, confirm the growth rate calculation:

\( r=e^{0.026}-1\approx1.02635 - 1 = 0.02635\approx2.64\% \) (or more precisely, using the formula \( r=k+\frac{k^{2}}{2}+\frac{k^{3}}{6}+\cdots \), for \( k = 0.026 \), \( r\approx0.026+\frac{(0.026)^{2}}{2}+\frac{(0.026)^{3}}{6}\approx0.026 + 0.000338+0.00000297\approx0.02634 \), so approximately \( 2.63\% \) or \( 2.6\% \) (but the exact value from \( e^{0.026}-1\approx0.02635 \), so about \( 2.64\% \))

Answer:

Country 4 has the greatest growth rate. The population of Country 4 is increasing by approximately \( 2.63\% \) (or \( 2.6\% \) or \( 2.64\% \)) each year. The country with the greatest growth rate is \(\boldsymbol{4}\).